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black rectangle black down-pointing triangle ... upper half inverse white circle square with lower right diagonal half black ...
Rectangle; Rhomboid; Rhombus; Square (regular quadrilateral) ... Incircle and excircles of a triangle; Nine-point circle; Circular sector; Circular segment; Crescent;
This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: adjacent angles in a parallelogram are supplementary (add to 180°) and,
Regular polygons; Description Figure Second moment of area Comment A filled regular (equiliteral) triangle with a side length of a = = [6] The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.
However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square.
An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
Pascal triangle; Peano curve; Penrose tiling; Pinwheel tiling; Pythagoras tree; Rauzy fractal; Rössler attractor; Sierpiński arrowhead curve; Sierpinski carpet; Sierpiński curve; Sierpinski triangle; Smith–Volterra–Cantor set; T-square; Takagi or Blancmange curve; Triflake [citation needed] Vicsek fractal; von Koch curve; Weierstrass ...
The book provided illustrated proof for the Pythagorean theorem, [31] contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon, the circle and square, as well as measurements of heights and distances. [32]